I have a very silly and basic question about finding charts for a manifold. The point is: I'm self learning differential geometry, however, I didn't find the answer for this in the book nor on the web. I've once asked about how to find charts, but now my point is another: is how to represent the elements of a manifold before start creating the charts.
First, I'm using Do Carmo's definition of manifold: A smooth manifold of dimension $n$ is a set $M$ with a family of bijective maps $\varphi_\alpha : U_\alpha \to M$ from open sets $U_\alpha\subset \mathbb{R}^n$ to $M$ such that:
- $\bigcup_\alpha\varphi_\alpha(U_\alpha)=M$
- For each pair $\alpha, \beta$ with $\varphi_\alpha(U_\alpha)\cap\varphi_\beta(U_\beta)=W\neq\emptyset$ we have $\varphi_\alpha^{-1}(W)$, $\varphi_\beta^{-1}(W)$ open in $\mathbb{R}^n$ and $\varphi_\beta^{-1}\circ\varphi_\alpha$, $\varphi_\alpha^{-1}\circ\varphi_\beta$ are differentiable.
- The family $\left\{U_\alpha, \varphi_\alpha\right\}$ is maximum with respect to conditions 1 and 2.
Amongst all definitions of smooth manifold, this one was the one I prefered to work with. My point here is: we make this definition in order to avoid the need to consider manifolds as subsets of some euclidean space. In other words, we want to deal with them without making reference to some ambient space.
My problem with this is to find the charts. I have to construct bijective functions from $\mathbb{R}^n$ to $M$, and so my doubt is: how I describe the elements of $M$?
The classical example of finding charts for the sphere $S^n$ assumes that the sphere is defined as a subset of $\mathbb{R}^{n+1}$, so we know that if $p \in S^n$ then there are $n$ real numbers $p^i$ such that $p = \left(p^1 ,\cdots, p^n\right)$ that simply satisfy some conditions. Then it becomes easier to find the charts because first of all we know how to describe the elements of the set. Second, because we can use the ambient space, so we can use stereographic projections, which depends on the ambient space.
But in general, we don't want to use one ambient space. So, if I was asked for instance to find the charts for the sphere without the ambient space, what should I do? Well, now if $p \in S^n$, I cannot say that $p$ is one $n$-tuple of numbers, and I cannot also use things from outside $S^n$ like the planes and lines used in stereographic projections.
I'm confused with all of that. In understand the theorems, the proofs, the use of transition charts to ensure differentiability, and so on. My only problem is to find the charts, I feel I'm in need of examples, but I haven't found many. Do Carmo's examples deal just with surfaces, and he always presents them as subsets of $\mathbb{R}^3$.
Can someone explain those points or point me some references? Sorry for such a silly and basic question. And also sorry for the long text, I just didn't find a way to make it smaller.
Best Answer
I hope that the following example is close to what you would like to see.
Let $ n \in \mathbb{N} $, and define an equivalence relation $ \sim $ on $ \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \} $ as follows: $$ \forall \mathbf{x}_{1},\mathbf{x}_{2} \in \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \}: \quad \mathbf{x}_{1} \sim \mathbf{x}_{2} \stackrel{\text{def}}{\iff} (\exists \lambda \in \mathbb{R} \setminus \{ 0 \}) (\mathbf{x}_{1} = \lambda \cdot \mathbf{x}_{2}). $$ Given any $ (x_{1},\ldots,x_{n+1}) \in \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \} $, we denote its $ \sim $-equivalence class by $$ [x_{1}:\ldots:x_{n+1}]. $$ We call $ (\mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \})/\sim $ the real projective $ n $-space, and we usually denote it by $ \mathbb{R} \mathbb{P}^{n} $. Intuitively, one can think of $ \mathbb{R} \mathbb{P}^{n} $ as the set of straight lines in $ \mathbb{R}^{n+1} $ that pass through the origin.
Observe that $ \mathbb{R} \mathbb{P}^{n} $ was not born as a subset of some ambient Euclidean space $ \mathbb{R}^{N} $. Although the elements of $ \mathbb{R} \mathbb{P}^{n} $ can be visualized as straight lines in $ \mathbb{R}^{n+1} $, this visualization is irrelevant if we are to treat the elements as points of an abstract space. Hence, at the most fundamental level, we should view $ \mathbb{R} \mathbb{P}^{n} $ as simply a set equipped with an equivalence relation, without worrying over how it can be embedded into Euclidean space.
Despite the abstract nature of $ \mathbb{R} \mathbb{P}^{n} $, we can, curiously enough, endow it with a manifold structure. For each $ i \in \{ 1,\ldots,n + 1 \} $, define a subset $ U_{i} $ of $ \mathbb{R} \mathbb{P}^{n} $ as follows: $$ U_{i} := \{ [x_{1}:\ldots:x_{n+1}] ~|~ x_{i} \neq 0 \}. $$ Next, define ‘chart’ maps $ \varphi_{i}: U_{i} \to \mathbb{R}^{n} $ by $$ \varphi([x_{1}:\ldots:x_{n+1}]) \stackrel{\text{def}}{=} \left( \frac{x_{0}}{x_{i}},\ldots,\widehat{\frac{x_{i}}{x_{i}}},\ldots,\frac{x_{n+1}}{x_{i}} \right) \in \mathbb{R}^{n}, $$ where the $ ~ \widehat{\hspace{4mm}} ~ $-symbol indicates an omitted term. Then $ \{ (U_{i},\varphi_{i}) \}_{i=1}^{n+1} $ is an atlas that makes $ \mathbb{R} \mathbb{P}^{n} $ an $ n $-dimensional manifold. We shall leave the derivation of the transition maps as an exercise.